kform: k-forms
In wedge: The Exterior Calculus

Description Usage Arguments Details Note Author(s) References See Also Examples

Functionality for dealing with k-forms

kform(S)
as.kform(M,coeffs,lose=TRUE)
kform_basis(n, k)
kform_general(W,k,coeffs,lose=TRUE)
## S3 method for class 'kform'
as.function(x,...)

`n`	Dimension of the vector space V=R^n
`k`	A k-form maps V^k to R
`W`	Integer vector of dimensions
`M`	Index matrix for a k-form
`coeffs`	Coefficients of the k-form
`S`	Object of class `spray`
`lose`	Boolean, with default `TRUE` meaning to coerce a 0-form to a scalar and `FALSE` meaning to return the formal 0-form
`x`	Object of class `kform`
`...`	Further arguments, currently ignored

A k-form is an alternating k-tensor.

Recall that a k-tensor is a multilinear map from V^k to the reals, where V=R^n is a vector space. A multilinear k-tensor T is alternating if it satisfies

omitted; see PDF

Function kform_basis() is a low-level helper function that returns a matrix whose rows constitute a basis for the vector space L^k(R^n) of k-tensors:

omitted; see PDF

and in fact

omitted; see PDF

where e_j,1<=j<=k is a basis for V.

In the wedge package, k-forms are represented as sparse arrays (spray objects), but with a class of c("kform", "spray"). The constructor function (kform()) ensures that rows of the index matrix are strictly nonnegative integers, have no repeated entries, and are strictly increasing.

Hubbard and Hubbard use the term “k-form”, but Spivak does not.

Robin K. S. Hankin

Hubbard and Hubbard; Spivak

ktensor,lose

as.kform(cbind(1:5,2:6),rnorm(5))
kform_general(1:4,2,coeffs=1:6)  # used in electromagnetism

K1 <- as.kform(cbind(1:5,2:6),rnorm(5))
K2 <- kform_general(5:8,2,1:6)
wedge(K1,K2)


f <- as.function(wedge(K1,K2))
E <- matrix(rnorm(32),8,4)

f(E) + f(E[,c(1,3,2,4)])  # should be zero